4.4.5 Distance Between a Point and a Line

This is computationally intense, so it typically appears in the last column.

The Formula

For line $ax + by + c = 0$ and point $(x_0, y_0)$:

$$\text{Distance} = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}$$

Example

Problem: Distance between point $(3, 1)$ and line $3x + 4y = -2$

Solution: Rewrite as $3x + 4y + 2 = 0$

$$d = \frac{|3(3) + 4(1) + 2|}{\sqrt{3^2 + 4^2}} = \frac{|9 + 4 + 2|}{\sqrt{25}} = \frac{15}{5} = 3$$

Tip: The line equation is usually a Pythagorean triple (3-4-5, 5-12-13) to make the denominator easy!

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Problem Set 4.4.5

Practice these problems. Type your answer and press Enter to check: